A heat flow for the mean field equation on a finite graph

Inspired by works of Castéras (Pac J Math 276:321–345, 2015), Li and Zhu (Calc Var Partial Differ Equ 58:1–18, 2019), Sun 60:1–26, 2021), we propose a heat flow for the mean field equation on connected finite graph $$G=(V,E)$$ . Namely $$\begin{aligned} \left\{ \begin{array}{l} \partial _t\phi (u)=\Delta u-Q+\rho \frac{e^u}{\int _Ve^ud\mu }\\ u(\cdot ,0)=u_0, \end{array}\right. \end{aligned}$$ where $$\Delta $$ is standard Laplacian, $$\rho real number, $$Q:V\rightarrow {\mathbb {R}}$$ function satisfying $$\int _VQd\mu =\rho , $$\phi :{\mathbb {R}}\rightarrow one certain smooth functions including (s)=e^s$$ We prove that any initial data $$u_0$$ \in there exists unique solution $$u:V\times [0,+\infty )\rightarrow above flow; moreover, u(x, t) converges to some $$u_\infty :V\rightarrow uniformly in $$x\in V$$ as $$t\rightarrow +\infty \Delta u_\infty -Q+\rho \frac{e^{u_\infty }}{\int _Ve^{u_\infty }d\mu }=0. Though G graph, this result still unexpected, even special case $$Q\equiv 0$$ Our approach reads follows: short time existence follows from ODE theory; various integral estimates give its long existence; moreover establish Lojasiewicz–Simon type inequality use it conclude convergence flow.